Astronomia nova

The Astronomia nova (full title in original Latin: Astronomia Nova ΑΙΤΙΟΛΟΓΗΤΟΣ seu physica coelestis, tradita commentariis de motibus stellae Martis ex observationibus G.V. Tychonis Brahe[1]) is a book, published in 1609, that contains the results of the astronomer Johannes Kepler's ten-year-long investigation of the motion of Mars. One of the greatest books on astronomy, the Astronomia nova provided strong arguments for heliocentrism and contributed valuable insight into the movement of the planets, including the first mention of their elliptical path and the change of their movement to the movement of free floating bodies as opposed to objects on rotating spheres. It is recognized as one of the most important works of the Scientific Revolution.[2]

Contents

Prior to Kepler, Nicolaus Copernicus proposed in 1543 that the Earth and other planets orbit the Sun. The Copernican model of the solar system was regarded as a device to explain the observed positions of the planets rather than a physical description.

Kepler sought for and proposed physical causes for planetary motion. His work is primarily based on the research of his mentor, Tycho Brahe. The two, though close in their work, had a tumultuous relationship. Regardless, on his deathbed, Brahe asked Kepler to make sure that he did not “die in vain,” and to continue the development of his Tychonic system. Kepler would instead write the Astronomia nova, in which he rejects the Tychonic system, as well as the Ptolemaic system and the Copernican system. Some scholars have speculated that Kepler’s dislike for Brahe may have had a hand in his rejection of the Tychonic system and formation of a new one.[3]

In English, the full title of his work is the New Astronomy, Based upon Causes, or Celestial Physics, Treated by Means of Commentaries on the Motions of the Star Mars, from the Observations of Tycho Brahe, Gent. For over 650 pages, Kepler walks his readers, step by step, through his process of discovery so as to dispel any impression of "cultivating novelty," he says.

The Astronomia nova's introduction, specifically the discussion of scripture, was the most widely distributed of Kepler’s works in the seventeenth century.[4] The introduction outlines the four steps Kepler took during his research. The first is his claim that the sun itself and not any imaginary point near the sun (as in the Copernican system) is the point where all the planes of the eccentrics of the planets intersect, or the center of the orbits of the planets. The second step consists of Kepler placing the sun as the center and mover of the other planets. This step also contains Kepler’s reply to objections against placing the sun at the center of the universe, including objections based on scripture. In reply to scripture, he argues that it is not meant to claim physical dogma, and the content should be taken spiritually. In the third step, he posits that the sun is the source of the motion of all planets, using Brahe’s proof based on comets that planets do not rotate on orbs. The fourth step consists of describing the path of planets as not a circle, but an oval.

As the Astronomia nova proper starts, Kepler demonstrates that the Tychonic, Ptolemaic, and Copernican systems are indistinguishable on the basis of observations alone. The three models predict the same positions for the planets in the near term, although they diverge from historical observations, and fail in their ability to predict future planetary positions by a small, though absolutely measurable amount. Kepler here introduces his famous diagram of the movement of Mars in relation to Earth if Earth remained unmoving at the center of its orbit. The diagram shows that Mars’s orbit would be completely imperfect and never follow along the same path.

Kepler discusses all his work at great length throughout the book. He addresses this length in the sixteenth chapter:

If thou art bored with this wearisome method of calculation, take pity on me, who had to go through with at least seventy repetitions of it, at a very great loss of time.[5]

Kepler, in a very important step, also questions the assumption that the planets move around the center of their orbit at a uniform rate. He finds that computing critical measurements based upon the Sun's actual position in the sky, instead of the Sun's "mean" position injects a significant degree of uncertainty into the models, opening the path for further investigations. The idea that the planets do not move in a uniform rate, but with a speed that varies their distance from the Sun, was completely revolutionary, and would become his second law (discovered before his first). Kepler, in his calculations leading to his second law, made multiple mathematical errors, which luckily cancelled each other out “as if by miracle.”[6]

Given this second law, he puts forth in Chapter 33 that the sun is the engine that moves the planets. To describe the motion of the planets, he claims the sun emits a physical species, analogous to the light it also emits, which pushes the planets along. He also suggests a second force within every planet itself that pulls it toward then sun to keep it from spiraling off into space.

Kepler then attempts to finally find the true path of the planets, which he determines is an ellipse. His initial attempt to define the orbit of Mars, far before he arrived at the ellipse shape, was off by only eight minutes, yet this was enough for Kepler to require an entirely new system. Kepler tried a number of shapes before the ellipse, including an egg shape. What’s more, he discovered the mathematical definition for the ellipse as the orbit, then rejected it, then adopted the ellipse without knowing that it was the same:

”I laid [the original equation] aside, and fell back on ellipses, believing that this was quite a different hypothesis, whereas the two, as I shall prove In the next chapter, are one in the same…Ah, what a foolish bird I have been!”[7]

That the speed of the planet changes at each moment such that the time between two positions is always proportional to the area swept out on the orbit between these positions.[9]

Kepler discovered the "second law" before the first. He presented his second law in two different forms: In Chapter 32 he states that the speed of the planet varies inversely based upon its distance from the Sun, and therefore he could measure changes in position of the planet by adding up all the distance measures, or looking at the area along an orbital arc. This is his so-called "distance law". In Chapter 59, he states that a radius from the Sun to a planet sweeps out equal areas in equal times. This is his so-called "area law".

However, Kepler's "area-time principle" did not facilitate easy calculation of planetary positions. Kepler could divide up the orbit into an arbitrary number of parts, compute the planet's position for each one of these, and then refer all questions to a table, but he could not determine the position of the planet at each and every individual moment because the speed of the planet was always changing. This paradox, referred to as the "Kepler problem," prompted the development of calculus.

Kepler discovered his "third law" a decade after the publication of the Astronomia nova as a result of his investigations in the 1619 Harmonices Mundi (Harmonies of the world).[10] He found that the ratio of the length of the semi-major axis of each planet's orbit (cubed), to the time of its orbital period (squared), is the same for all planets.

In his introductory discussion of a moving earth, Kepler addressed the question of how the Earth could hold its parts together if it moved away from the center of the universe which, according to Aristotelian physics, was the place toward which all heavy bodies naturally moved. Kepler proposed an attractive force similar to magnetism, which may have been known by Newton.

"Gravity is a mutual corporeal disposition among kindred bodies to unite or join together; thus the earth attracts a stone much more than the stone seeks the earth. (The magnetic faculty is another example of this sort).... If two stones were set near one another in some place in the world outside the sphere of influence of a third kindred body, these stones, like two magnetic bodies, would come together in an intermediate place, each approaching the other by a space proportional to the bulk [moles] of the other.... For it follows that if the earth's power of attraction will be much more likely to extend to the moon and far beyond, and accordingly, that nothing that consists to any extent whatever of terrestrial material, carried up on high, ever escapes the grasp of this mighty power of attraction.”[11]

Kepler discusses the moon's gravitational effect upon the tides as follows:[12][13]

The sphere of the attractive virtue which is in the moon extends as far as the earth, and entices up the waters; but as the moon flies rapidly across the zenith, and the waters cannot follow so quickly, a flow of the ocean is occasioned in the torrid zone towards the westward. If the attractive virtue of the moon extends as far as the earth, it follows with greater reason that the attractive virtue of the earth extends as far as the moon and much farther; and, in short, nothing which consists of earthly substance anyhow constituted although thrown up to any height, can ever escape the powerful operation of this attractive virtue.

Johannes also clarifies the concept of lightness in terms of relative density, in opposition to the Aristotelian concept of the absolute nature or quality of lightness as follows. His argument could easily be applied today to something like the flight of a hot air balloon.

Nothing which consists of corporeal matter is absolutely light, but that is comparatively lighter which is rarer, either by its own nature, or by accidental heat. And it is not to be thought that light bodies are escaping to the surface of the universe while they are carried upwards, or that they are not attracted by the earth. They are attracted, but in a less degree, and so are driven outwards by the heavy bodies; which being done, they stop, and are kept by the earth in their own place.[14]

In reference to Kepler's discussion relating to gravitation, Walter William Bryant makes the following statement in his book Kepler (1920).

...the Introduction to Kepler’s “Commentaries on the Motion of Mars,” always regarded as his most valuable work, must have been known to Newton, so that no such incident as the fall of an apple was required to provide a necessary and sufficient explanation of the genesis of his Theory of Universal Gravitation. Kepler’s glimpse at such a theory could have been no more than a glimpse, for he went no further with it. This seems a pity, as it is far less fanciful than many of his ideas, though not free from the “virtues” and “animal faculties,” that correspond to Gilbert’s “spirits and humours”.[15]

Kepler considered that this attraction was mutual and was proportional to the bulk of the bodies, but he considered it to have a limited range and he did not consider whether or how this force may have varied with distance. Furthermore, this attraction only acted between "kindred bodies"—bodies of a similar nature, a nature which he did not clearly define.[16][17] Kepler's idea differed significantly from Newton's later concept of gravitation and it can be "better thought of as an episode in the struggle for heliocentrism than as a step toward Universal gravitation.[18]

^In his Astronomia nova, Kepler presented only a proof that Mars' orbit is elliptical. Evidence that the other known planets' orbits are elliptical was presented later.
See: Johannes Kepler, Astronomia nova … (1609), p. 285. After having rejected circular and oval orbits, Kepler concluded that Mars' orbit must be elliptical. From the top of page 285: "Ergo ellipsis est Planetæ iter; … " (Thus, an ellipse is the planet's [i.e., Mars'] path; … ) Later on the same page: " … ut sequenti capite patescet: ubi simul etiam demonstrabitur, nullam Planetæ relinqui figuram Orbitæ, præterquam perfecte ellipticam; … " ( … as will be revealed in the next chapter: where it will also then be proved that any figure of the planet's orbit must be relinquished, except a perfect ellipse; … ) And then: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , be a perfect ellipse: … ) The geometric proof that Mars' orbit is an ellipse appears as Protheorema XI on pages 289-290.
Kepler stated that all planets travel in elliptical orbits having the Sun at one focus in: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, part 1, III. De Figura Orbitæ (III. On the figure [i.e., shape] of orbits), pages 658-665. From p. 658: "Ellipsin fieri orbitam planetæ … " (Of an ellipse is made a planet's orbit … ). From p. 659: " … Sole (Foco altero huius ellipsis) … " ( … the Sun (the other focus of this ellipse) … ).

^In his Astronomia nova … (1609), Kepler did not present his second law in its modern form. He did that only in his Epitome of 1621. Furthermore, in 1609, he presented his second law in two different forms, which scholars call the "distance law" and the "area law".

His "distance law" is presented in: "Caput XXXII. Virtutem quam Planetam movet in circulum attenuari cum discessu a fonte." (Chapter 32. The force that moves a planet circularly weakens with distance from the source.) See: Johannes Kepler, Astronomia nova … (1609), pp. 165-167.On page 167, Kepler states: " … , quanto longior est αδ quam αε, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud δ, quam in æquali arcu excentrici apud ε." ( … , as αδ is longer than αε, so much longer will a planet remain on a certain arc of the eccentric near δ than on an equal arc of the eccentric near ε.) That is, the farther a planet is from the Sun (at the point α), the slower it moves along its orbit, so a radius from the Sun to a planet passes through equal areas in equal times. However, as Kepler presented it, his argument is accurate only for circles, not ellipses.

His "area law" is presented in: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ), Protheorema XIV and XV, pp. 291-295. On the top p. 294, it reads: "Arcum ellipseos, cujus moras metitur area AKN, debere terminari in LK, ut sit AM." (The arc of the ellipse, of which the duration is delimited [i.e., measured] by the area AKM, should be terminated in LK, so that it [i.e., the arc] is AM.) In other words, the time that Mars requires to move along an arc AM of its elliptical orbit is measured by the area of the segment AMN of the ellipse (where N is the position of the Sun), which in turn is proportional to the section AKN of the circle that encircles the ellipse and that is tangent to it. Therefore, the area AMN that is swept out by a radius from the Sun to Mars as Mars moves along an arc AM of its elliptical orbit is proportional to the time that Mars requires to move along that arc. Thus, a radius from the Sun to Mars sweeps out equal areas in equal times.

In 1621, Kepler restated his second law for any planet: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, page 668. From page 668: "Dictum quidem est in superioribus, divisa orbita in particulas minutissimas æquales: accrescete iis moras planetæ per eas, in proportione intervallorum inter eas & Solem." (It has been said above that, if the orbit of the planet is divided into the smallest equal parts, the times of the planet in them increase in the ratio of the distances between them and the sun.) That is, a planet's speed along its orbit is inversely proportional to its distance from the Sun. (The remainder of the paragraph makes clear that Kepler was referring to what is now called angular velocity.)

^Johannes Kepler, Harmonices Mundi [The Harmony of the World] (Linz, (Austria): Johann Planck, 1619), p. 189. From the bottom of p. 189: "Sed res est certissima exactissimaque quod proportio qua est inter binorum quorumcunque Planetarum tempora periodica, sit præcise sesquialtera proportionis mediarum distantiarum, … " (But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialternate proportion [i.e., the ratio of 3:2] of their mean distances, … ")
An English translation of Kepler's Harmonices Mundi is available as: Johannes Kepler with E.J. Aiton, A.M. Duncan, and J.V. Field, trans., The Harmony of the World (Philadelphia, Pennsylvania: American Philosophical Society, 1997); see especially p. 411.